Probability Estimation in the Framework of Intuitionistic Fuzzy Evidence Theory

Mathematical Problems in Engineering, Apr 2015

Intuitionistic fuzzy (IF) evidence theory, as an extension of Dempster-Shafer theory of evidence to the intuitionistic fuzzy environment, is exploited to process imprecise and vague information. Since its inception, much interest has been concentrated on IF evidence theory. Many works on the belief functions in IF information systems have appeared. Although belief functions on the IF sets can deal with uncertainty and vagueness well, it is not convenient for decision making. This paper addresses the issue of probability estimation in the framework of IF evidence theory with the hope of making rational decision. Background knowledge about evidence theory, fuzzy set, and IF set is firstly reviewed, followed by introduction of IF evidence theory. Axiomatic properties of probability distribution are then proposed to assist our interpretation. Finally, probability estimations based on fuzzy and IF belief functions together with their proofs are presented. It is verified that the probability estimation method based on IF belief functions is also potentially applicable to classical evidence theory and fuzzy evidence theory. Moreover, IF belief functions can be combined in a convenient way once they are transformed to interval-valued possibilities.

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Probability Estimation in the Framework of Intuitionistic Fuzzy Evidence Theory

Probability Estimation in the Framework of Intuitionistic Fuzzy Evidence Theory Yafei Song and Xiaodan Wang Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China Received 10 November 2014; Revised 24 March 2015; Accepted 9 April 2015 Academic Editor: Joao B. R. Do Val Copyright © 2015 Yafei Song and Xiaodan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Intuitionistic fuzzy (IF) evidence theory, as an extension of Dempster-Shafer theory of evidence to the intuitionistic fuzzy environment, is exploited to process imprecise and vague information. Since its inception, much interest has been concentrated on IF evidence theory. Many works on the belief functions in IF information systems have appeared. Although belief functions on the IF sets can deal with uncertainty and vagueness well, it is not convenient for decision making. This paper addresses the issue of probability estimation in the framework of IF evidence theory with the hope of making rational decision. Background knowledge about evidence theory, fuzzy set, and IF set is firstly reviewed, followed by introduction of IF evidence theory. Axiomatic properties of probability distribution are then proposed to assist our interpretation. Finally, probability estimations based on fuzzy and IF belief functions together with their proofs are presented. It is verified that the probability estimation method based on IF belief functions is also potentially applicable to classical evidence theory and fuzzy evidence theory. Moreover, IF belief functions can be combined in a convenient way once they are transformed to interval-valued possibilities. 1. Introduction The Dempster-Shafer theory of evidence, also called belief function theory, is an important method to deal with uncertainty in information systems. Since it was firstly presented by Dempster [1], and was later extended and refined by Shafer [2], the Dempster-Shafer theory, or the D-S theory for short, has generated considerable interest. Its application has extended to many areas such as expert systems [3], fault reasoning [4, 5], pattern classification [6–9], information fusion [10], knowledge reduction [11], global positioning system [12], regression analysis [13], and data mining [14]. The theory of fuzzy set, proposed by Zadeh [15], is another mathematical tool for handling uncertainty. It has received a great deal of attention due to its capability in uncertainty reasoning. Therefore, over the last decades, several generalizations of fuzzy set have been introduced in the literature. Intuitionistic fuzzy (IF) set proposed by Atanassov [16] is one of the generalizations of fuzzy set which is capable of dealing with vagueness much better. A fuzzy set only gives a membership degree to describe an element belonging to a set, while an intuitionistic fuzzy set gives both a membership degree and a nonmembership degree. Thus, an IF set is more objective than a fuzzy set to describe the vagueness of data. As a fuzzy set can be reviewed as a fuzzy event, an IF set is also an IF event. Relationship between fuzzy set theory and belief function theory has been focused on for a long time. Zadeh was the first to generalize the Dempster-Shafer theory to fuzzy sets, based on his work on the concept of information granularity and the theory of possibility [17, 18]. He suggested how to compute probabilities of fuzzy events and showed some basic properties of probabilities of fuzzy events [19]. Following Zadeh’s work, Ishizuka, Yager, Ogawa, John Yen, and Zhu have extended the D-S theory to fuzzy sets in different ways [20–24]. As an inception, belief functions on IF events were investigated by Grzegorzewski and Mrówka in [25], where basic properties of probability measures for IF events were studied. Riečan gave an axiomatic characterization of a probability on IF events in [26] and proved a representation theorem for it in [27]. In [28, 29], Gerstenkorn and Mańko gave two new definitions of the IF probability: the first probability of an IF set is a real number in using the integral operation, and the second probability of an IF set is also an IF set based on the level sets. In [30], Gerstenkorn and Mańko defined a probability of IF events, which is defined by the membership degree and half of the hesitancy margin of every element, where the probability of an IF set is a real number. Feng et al. [31] proposed a novel pair of belief and plausibility functions defined by employing intuitionistic fuzzy lower and upper approximation operators. All the above works on IF belief function theory focused on the determination of the basic probabilities assigned to IF events based on the known probabilistic distribution in the universe of discourse. However, its converse problem, that is, probability estima (...truncated)


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Yafei Song, Xiaodan Wang. Probability Estimation in the Framework of Intuitionistic Fuzzy Evidence Theory, Mathematical Problems in Engineering, 2015, 2015, DOI: 10.1155/2015/412045